/var/tmp is the default location for Audacity’s temporary folder.
Running Audacity should create a folder:
Audacity must be able to create that folder (and write to it).
Right click on /var/tmp and select “Properties” then open the “Permissions” tab. The Permissions will probably be greyed out (because the folder is owned by “root”) but should say that “Others” have permission to “Create and delete files”.
if this turns out to be the problem then you can either change the permissions of that folder (do that as root), or manually change the Audacity preferences (~/.audacity-data/audacity.cfg) to specify a different location for the temp folder.
The problem is that your /tmp is mounted with no file system. I would change that none to tmpfs…
There is a lot more going on in the way of writes in the /var directory than just the logs.
/run/user/1000 is already mounted as tmpfs if /run is mounted as tmpfs.